September 2008 - mathfun@mailman.xmission.com

Re: [math-fun] Equation mixing integers and reals
by Henry Baker 16 Sep '08

16 Sep '08

Check out Gomory's Cutting Plane Method for integer linear programming: http://en.wikipedia.org/wiki/Cutting-plane_method You may not be able to tell from the description above, but if you do this in 2 dimensions, you will find that you are doing the same calculation as a continued fraction. At 10:44 AM 9/16/2008, Christian Boyer wrote: >Bonjour Henry, > >Ooops, sorry, I saw your answer after those from Dan and Victor. >Yes, my problem is close to these other problems. > >Christian. > >-----Message d'origine----- >De : Henry Baker [mailto:hbaker1@pipeline.com] >EnvoyÃ© : mardi 16 septembre 2008 16:42 >Ã : Christian Boyer >Cc : 'math-fun' >Objet : Re: [math-fun] Equation mixing integers and reals > >Isn't this similar to the "integer linear programming" problem (integer >version of linear programming problem) ? > >It looks like you may be able to reduce this to some sort of knapsack >problem. > >Of course, there's always the issue of how to represent the "reals" in this >problem. > >At 06:42 AM 9/16/2008, Christian Boyer wrote: >>Let f(x , x , ..., x ) = k x + k x + ... + k x >> 1 2 n 1 1 2 2 n n >> >> where k , k , ..., k are n REAL constants >> 1 2 n >> >> and x , x , ..., x are n INTEGERS (positive or negative) >> 1 2 n >> >>If we try to have the best possible approximations of >> >> f(x , x , ... x ) = 0 >> 1 2 n >> >>do you know good algorithms giving >>best values of (x , x , ..., x )? >> 1 2 n >> >>When n = 2, a good algorithm is to compute the >>continued fraction of k / k >> 2 1 >>At each step of the algorithm, the continued fraction >>will give more and more excellent (x , x ). >> 1 2 >> >>But which algorithm(s) for n > 2? >> >>Christian.

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Re: [math-fun] Mersenne Indexes Revealed
by Dan Asimov 16 Sep '08

16 Sep '08

Guy wrote: << Now it can be told. In order of discovery, M(p) with p = 43,112,609 or 37,156,667 is prime. UCLA discovered the first and win the $100,000 EFF prize for the first known prime of over 10m decimal digits. The second prime was discovered two weeks later, so they unfortunately miss out on the prize. >> Actually, at the GIMPS website, they write that of the $100,000, $50,000 is going to the UCLA math dept., but $25,000 will be distributed to the main discoverers of the last 6 Mersenne primes. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] Mersenne Indexes Revealed
by Guy Haworth 16 Sep '08

16 Sep '08

Now it can be told. In order of discovery, M(p) with p = 43,112,609 or 37,156,667 is prime. UCLA discovered the first and win the $100,000 EFF prize for the first known prime of over 10m decimal digits. The second prime was discovered two weeks later, so they unfortunately miss out on the prize. www.mersenne.org Guy

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[math-fun] Bathtub drains
by Dan Asimov 16 Sep '08

16 Sep '08

Long ago I read that water draining from a bathtub swirls one way in the N. hemisphere and the opposite way in the S. hemisphere. (Something about Coriolis forces.) But since then I've heard a number of times that this is just a myth. Does anyone know which is really true? (Perhaps it depends on the precision of your equipment?) --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] New Mersenne Primes: still no indexes
by Guy Haworth 16 Sep '08

16 Sep '08

Still no indexes, but two new prime Mp have been multiply confirmed. The longer the pregnant pause, the more likely that one is the first known prime of over 10m decimal digits ... www.mersenne.org Guy [MM31 remains the largest Double Mersenne Number whose primality status is known :-)]

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[math-fun] Almost integer cuboid problem
by David Wilson 16 Sep '08

16 Sep '08

Let an almost integer cuboid (rectangular box with rectangular edges and face diagonals) have edges a, b, c. What is the largest integer necessarily dividing abc?

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[math-fun] Doin' the Hilbert walk
by Fred lunnon 15 Sep '08

15 Sep '08

I've been spending quality time with Hilbert walks (finite approximations to Hilbert curves) in d-space lately (don't tell RWG), my ruminations somewhat inconvenienced by inability to locate a definition of what might actually constitute a Hilbert walk/curve in any dimension d > 2. Previous authors seem to cheerfully ignore the difficulty; or perhaps they just implicitly define a walk as being whatever convoluted Hmaitonian path their proposed algorithm happens to construct. While I have some ideas about an appropriate general definition, it's still not very clear to me exactly what properties --- combinatorial or possibly staistical --- of such a path might be significant in applications --- such as multiple-key database, and equation solving in many variables. Can anybody out there cast light on these matters? Fred Lunnon

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Re: [math-fun] Bathtub drains
by Dan Asimov 15 Sep '08

15 Sep '08

Simon wrote: << Yes, this is an urban legend, actualy some people tried it several times. >> Thanks for the references! But in fact, it is really an issue of the precision of the experiment, judging from the 1962 MIT experiment cited by Cecil Adams that Simon linked to. When it's done very very carefully, letting still water drain from a symmetrical vessel in a symmetrical manner *does* result in counterclockwise swirling in the Northern Hemisphere! (Indeed the sense of the swirl is likely to be random for a casually done home experiment in an actual bathtub, since the teeny Coriolis effect will almost surely be swamped by the asymmetry of the situation.) --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] Reflection from a moving mirror
by Dan Asimov 14 Sep '08

14 Sep '08

Re the moving mirror -- it's dawning on me that I've not been clear on the meaning of a mirror "moving parallel to itself". If this means moving so it *remains* parallel to itself. then there are many kinds of motion that fit that description. If it means moving so that it remains in the same plane, ditto. But the paper mentioned by Censor uses this phrase. Can someone explain to me what it means? Thanks, Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] Reflection from a moving mirror
by Eugene Salamin 14 Sep '08

14 Sep '08

----- Original Message ---- From: Mike Stay <metaweta(a)gmail.com> To: math-fun <math-fun(a)mailman.xmission.com> Sent: Thursday, September 4, 2008 12:59:51 PM Subject: Re: [math-fun] Reflection from a moving mirror Some references: http://www.nature.com/nature/journal/v248/n5450/abs/248660a0.html http://tinyurl.com/6zl9cs R. Gerharz, "Detection of the transverse Doppler effect with laser light." Proc. IEEE (Corresp.),vol. S2,p. 218, Feb. 1964. ----------------------------------------- I downloaded the Nature paper by Jennison and Davies. The experiment looked for a transverse Doppler shift from a rotating mirror, and it was claimed to be sensitive to 1 part in 10^16. A rotating mirror with a speed of 2-20 m/s formed one arm of a Michelson interferometer. A fixed mirror on a piezo stage formed the other arm, allowing the apparatus to be tested by introducing a first-order Doppler shift. A second-order Doppler shift of the expected magnitude would have produced a beat frequency of 1 to 1/12 Hz. No such beat was seen. I can email a copy of the 2-page paper to anyone who is interested. Gene

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